Team Review

• How did everyone feel about the problem set?

• Any questions?

• Discuss with Team

Disclaimer

Functions are like baking receipes

Properties of functions

For each combination of independent variables, there is exactly one value of the dependent variable.

• \(f(1)\ne\{2,3\}\) i.e. if I put 1 into the function I can’t have both 2 and 3 come out

• Vertical line test

Functions can be continuous or discontinuous

• Continuous: For every value of x, f(x) returns a number

• Discontinuous: Parts are undefined

Are these functions

Limits

Definition:

The value a function approaches as the input approaches a specific value

\[ \large \lim_{x\to c} f(x)=L \]

Verbally:

“The limit of the function f(x) as x approaches the value c is L”

Start with an example

• What is the limit of \(f(x)=2x^2-4\) as x approaches 2?

From both directions it looks like f(x) converges to 4

\[ \lim_{x\to c}(2x^2-4)=4 \]

Graphical Solution

What happens with discontinuous functions?

Try with your group the same approach for:

\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]

Hint: Start very close to 2 like 1.9 and 2.1

We broke the universe

Clearer on a graph

Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)

Defined Limit at an undefined point

• Discontinuous functions can still have limits

• Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2

Work with your team to complete these task

Work with your team to complete these task

2. Can you think of examples where discontinuous functions might exist in environmental science?

3. Choose a a team to draw an example of one of these statements:

• \(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite

• \(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)

• \(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)

• \(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)

Putting it all together

Recall average rate of change and instantaneous

Taking the average rate of change to a set limit will eventually converge to the instantaneous.

Walk Trhough Example

Walk Through Example

Walk Through Example

Walk Through Example

Walk Through Example

Walkthrough

Tangent Lines

What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).

These are called tangent lines

Put our example in math notation

\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]

Choose a point along the function \((x,f(x))\)

Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)

Add these into the slope equation

\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Derivative Definition

\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

\[ \large f'(x)\text{, or }\frac{dy}{dx} \]

Not all functions are differentiable

Calculus and Derivatives are the study of change

Environmental Science is also a study of change

Environmental Science is also a study of change

Environmental Science is also a study of change

Rules for Differentiation

• If \(f(x)\) is contant, then for all \(x\), \(f'(x)=0\)

\[ \begin{align} &y=a &\frac{dy}{dx}=0 \end{align} \]

\[ \frac{d}{dx}[x^n] =nx^{n-1} \]

Examples

\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]

Rules for Differentiation

\[ \begin{align} \frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\ \frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x) \end{align} \]

Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.

Example

\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]

1. As a team, list 5 fields of environmental science where studying the rate of change and derivatives would be important

2. Which rules should you use to take these derivatives?

\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]

3. Find the derivatives of these functions

\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]

4. Discuss why derivatives need continuity. It might help to think about why some continuous functions don’t have derivatives.